Options A and B are correct. If Katarina asks Dacia how she can derive the formula r= x²+y², then the following statements should be Dacia's reply.
- Because r is the hypotenuse of a right triangle, with legs x and y, you can use the Pythagorean theorem to determine 'r
- r is a straight line distance from the point and can be projected onto x and y axes where x = rcosθ, y = rsinθ. Writing sinθ = y/r, Cosθ = x/r and using the identity sin²θ+cos²θ = 1
Given the equation by Katarina expressed as:
r = √x²+y²
One of the ways Katarina can derive the equation is to use the Pythagoras theorem which states that the sum of the square of the hypotenuse is equal to the sum of the other two sides. Let:
r be the length of the hypotenuse
x and y be the other two sides
According to the theorem:
r² = x² + y²
Take the square root of both sides
√r² = √x² + y²
r = √x² + y²
Also, Katarina can use the trigonometry identity. According to a right triangle where:
sinθ = y/r
Cosθ = x/r
According to trigonometry identity, sin²θ+cos²θ = 1
(y/r)² + (x/r)² = 1
y²/r² + x²/r² = 1
(y²+x²)/r² = 1
y²+x² = r²
Swap
r²=y²+x²
This shows that the first two options are correct.
Because r is the hypotenuse of a right triangle, with legs x and y, you can use the Pythagorean theorem to determine 'r
r is a straight line distance from the point and can be projected onto x and y axes where x = rcosθ, y = rsinθ. Writing sinθ = y/r, Cosθ = x/r and using the identity sin²θ+cos²θ = 1
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